# A note on Lefschetz spheres and their relatives

## TL;DR

This paper explores specific classes of PL-spheres, particularly those that are both vertex-decomposable and strongly edge-decomposable, linking them to recent proof methods related to the $g$-conjecture.

## Contribution

It identifies a class of spheres where vertex-decomposability and strong edge-decomposability intersect, enabling the application of Adiprasito's proof techniques.

## Key findings

- Identified the intersection of vertex-decomposable and strongly edge-decomposable spheres.
- Linked this class to the applicability of Adiprasito's proof method.
- Provided insights into the structure of spheres related to the $g$-conjecture.

## Abstract

Inspired by the works of Adiprasito, Babson, Nevo, and Murai on the $g$-conjecture, we consider different classes of PL-spheres and the relations between them. We focus on a certain class of spheres that is in the intersection of vertex-decomposable spheres (a concept due to Provan and Billera) and strongly edge-decomposable spheres (a concept due to Eran Nevo). The spheres in this class are exactly those vertex-decomposable ones for which Adiprasito's recent proof method works.